Applied Mechanics

International Library of Technology

374B

Applied Mechanics and Strength of Materials 197 ILLUSTRATIONS

Prepared Under Supervision of

A. B.

CLEMENS

DIRECTOR, MECHANICAL SCHOOLS, INTER-

NATIONAL

CORRESPONDENCE

SCHOOLS

LINK MECHANISMS GEARING GEAR TRAINS AND CAMS PULLEYS AND BELTING MATERIALS OF CONSTRUCTION STRENGTH OF MATERIALS THE TESTING OF MATERIALS

Published by

INTERNATIONAL TEXTBOOK COMPANY SCRANTON, PA.

Link Mechanisms: Gearing:

Copyright. 1906, by INTERNATIONAL

Copyright,

(Pulleys

and Belting:

TEXTBOOK COMPANY.

by INTERNATIONAL TEXTBOOK COMPANY. Copyright, 1906, by INTERNATIONAL TEXTBOOK COMPANY.

1906,

Gear Trains and Cams:

1906,

Copyright,

Materials of Construction:

Copyright,

COMPANY. Strength of Materials, Parts

BOOK COMPANY. The Testing of Materials:

1

and

2:

Copyright,

by

INTERNATIONAL TEXTBOOK COMPANY.

1927 AJJQ6, by INTERNATIONAL

-**"-.*,.

TEXTBOOK

Copyright, 1906, by INTERNATIONAL TEXT1906,

by INTERNATIONAL TEXTBOOK COM-

PANY.

Entered

at Stationers'

Hall, London.

All rights reserved

Printed in U. S, A.

6095

sty'

PRUSS OF

INTERNATIONAL TEXTBOOK COMPANY SCRANTON, PA.

374B

94217

PREFACE The volumes of the International Library made up of Instruction Papers, or

of Technology are

Sections, comprising the various courses of instruction for students of the International

Correspondence Schools. The original manuscripts are prepared by persons thoroughly qualified both technically and by experience to write with authority, and in cases are ^

many

they

regularly employed elsewhere in practical work as experts. The manuscripts are then carefully edited to make them suitable for correspondence instruction. The Instruction Papers are written clearly and in the simplest language possible, so as to make them understood readily by all students. Necessary technical expressions are clearly explained when introduced.

The selves

more

great majority of our students wish to prepare themadvancement in their vocations or to qualify for

for

congenial occupations. Usually they are employed and able to devote only a few hours a day to study. Therefore every effort must be made to give them practical and accurate information in clear and concise form and to make this infor-

mation include essentials.

freely.

These

Illustrating

all

of

To make

the the

illustrations

Department

requirements of the

but none of the non-

essentials

text

clear,

illustrations

are especially

in order to

are

used

made by our own

adapt them fully to the

text.

In the table of contents that immediately follows are given the titles of the Sections included in this volume, and under eacli title are listed the main topics discussed.

INTERNATIONAL TEXTBOOK COMPANY B

CONTENTS NOTE. This volume is made up of a number of separate as indicated by their titles, and the page numbers of each usually this list of contents the titles of the parts are given in the order in in the book, and tinder each title is a full synopsis of the subjects

parts,

or sections,

begin with 1. In which they appear treated.

LINK MECHANISMS

Pages

Relative Motions of Links

1-35

Introduction

1-2 3-8

General Kinematic Principles Kinds of constrained motion; Plane motion of a rigid body; Relative motion.

9-35

Levers mechanism;

Steam-engine

Quick-return

motions;

Straight-line motion; Universal joint.

GEARING 1-46

Toothed Gearing Rolling Curves on Surfaces

1-4 5-32

Spur Gearing Involute systems Cycloidal, or principles rolled-curve, system; Proportions of gear-teeth; Construction of tooth profiles.

General

;

;

33-40 41-46

Bevel Gearing Spiral and

Worm-Gearing

GEAR TRAINS AND CAMS !~29 6-14

Gear Trains Engine-Lathe Gear Trains Back-gear train; Screw-cutting Epicyclic Trains

train.

15-23

CONTENTS

vi

GEAR TRAINS AND CAMS

(Continued)

Pages Revolving Gear Trains

24-25

Reversing Mechanisms Cams and Cam Trains

26-29 30-45

Rotary Cams Sliding and Cylindrical Cams Ratchet Mechanisms

40-41

31-39 42-45

PULLEYS AND BELTING Belt Gearing Kinematics of Flexible Gearing Length of Open and Crossed Belts

1-33 ,

1-5 6-17 18-22

Power Transmission by Belt Care and Use of Belting

23-26 27-33

Belt Connections for Non-Parallel Shafts

MATERIALS OF CONSTRUCTION 1-54

Metals Iron

1-20

Manufacture and Properties of Iron

1-7

iron; Ores of iron; Separation of iron ores; Blast furnace; Pig iron.

Forms of its

from

3-16

Cast Iron Cupola; Characteristics of cast iron; Carbon Silicon, sulphur, phosphorus, and manganese

in cast iron;

m

cast iron

;

Change of volume of cast iron Chilled castings Malleable castings;' Case-hardening malleable castings. ;

;

16-20

Wrought Iron Purity of wrought iron Puddling furnace Siemens regenerative furnace; Puddling and rolling; Properties of wrought iron; Defects of wrought iron. ;

;

21-42 21-22

Steel Classification

Steel

Open-Hearth Blister Steel

and Bessemer Steel

and Shear Steel

Crucible Steel Superiority of crucible steel

Electric Furnace Steel

;

Tool

23-26 27 28-30

steel.

30-31

CONTENTS MATERIALS OF .CONSTRUCTION

vii

(Continued)

Pages 32-34

Ingots and Steel Castings ......................... Ingots; Shaping of ingots; Defects in ingots; Steel castings.

35-42

Alloy Steels .................................... steel; Air-hardening, or self -hardening, steel; Manganese steel; Nickel steel; Chrome steel; Nickelchrome steel; Chrome-vanadium steel; Stainless steel;

Tungsten

Titanium

steel.

43-54 43-47

Non-Ferrous Alloys ............................. Brasses and Bronzes ............................. Value of non-ferrous alloys; Brass; Bronze.

48

Bronze and Brass Castings ................... Miscellaneous Alloys ............................ Copper-nickel alloys; German silver; Silicon bronze;

48-51

Copper-manganese alloy; Babbitt metal; Solder.

52

Aluminum Alloys ............................... Magnalium; Duralumin; Aluminum-zinc alloys; Aluminum-copper alloy. Special Alloys ..... ............................. '

53 ~ 54

Die-casting alloys; Stellite.

Non-Metals ..................................... Mortars and Concrete ............................

55-71

55-59

Portland cement; Portland-cement concrete; Portlandcement mortar; Difference between mortar and concrete Cement Cement paste Proportioning of concrete

;

;

;

;

Methods of mixing concrete.

Timber

..................................... ;

Characteristics

of

Evergreen timber;

timber;

Tropical

Hardwood timber Wood preservatives. Transmission Rope ............................... Manufacture of rope; Mounting of hemp rope. Selection of Materials ........................... timber

;

;

STRENGTH OF MATERIALS, PART

67-68 69 ~ 71

1

and Elasticity ................ Stress and Deformation .......................... Elasticity ......................................

Stress, Deformation,

Tension ........................................ Compression ....................................

1-18 1- 2 3~ 5

6-7

CONTENTS

viii

STRENGTH OF MATERIALS, PART

I

(Continued')

Pages 9-10

Shear Factors of Safety Pipes and Cylinders

11-13 14-18 19-26

Statics

Elementary Graphic

Force diagram and equilibrium polygon; Composition of moments; Graphic expressions for moments.

Beams

27-52

27

Definitions

Simple Beams

28-43

Reactions of supports; Vertical shear; Bending moment; Simple beam with uniform load; Simple beams with

mixed

loads.

Overhung Beams and

44-52

Cantilevers

STRENGTH OF MATERIALS, PART Beams

2

1_15

Strength of Beams Deflection of

1-10

Beams

11-13

Comparison of Strength and Columns Torsion and Shafts Ropes and Chains .

Stiffness of

Beams

14-15

16-22

23-28 29-34

.

THE TESTING OF MATERIALS Methods and Appliances

1_39

Test Pieces

2-4

Apparatus for Testing Materials Tension Test

5_14 15-17

Records of the Test

Compression Test Method of making Transverse Test

the test; Records of the

Shearing and Torsion Tests Miscellaneous Tests

18-24 25-27 test.

28-31 32

32-39

LINK MECHANISMS Serial 990

Edition

1

RELATIVE MOTIONS OF LINKS INTRODUCTION 1.

Mechanics.

The science that treats of the of bodies and of the forces that produce or tend to such motions is called mechanics.

motions produce

Applied mechanics comprises the principles of pure applied to the design and construction of machinery or to works of engineering The part of mechanics as

mechanics that relates to machinery of

machinery. The mechanics

chief branches:

of machinery

applied

is

called the

may be

mechanics

divided into two. treating of the

kinematics of machinery,

motions of machine parts without regard to the forces actingdynamics of machinery, treating of the forces acting on machine parts and of the transmission of force from one

part to another.

Free and Constrained Motion. A body is said to it may move in any direction in obedience to the forces acting on it. A body is constrained when the nature or direction of its motion is determined by its con2.

be free when

nection with other bodies.

Examples of free bodies are the moon, the sun, and other heavenly bodies. Examples of constrained m every machine. Thus, the crosshead of anbodies are seen engine is conSUldeS Md the Shaft the b

w

If

'

COPV R1S H TKD BV NTBRNAT10NAL TBXTB ,

^

-ings

in

LINK MECHANISMS In

body

may Any

constrained motion, every point of the constrained is forced to move in a definite path, no matter what IDC the direction of the force that causes the motion.

lorce that tends to give the body some other motion is once neutralized by an equal and opposite force developed For example, the block a, in the constraining members. Fig. 1, is enclosed by stationary guides b and c and its only possible motion therefore is a sliding motion along the line EF. Suppose that the acting force P has the direction of at

the line

M N.

Then,

if

P

a were free, the force would cause it to move

M the components H and N. along the line Let P be resolved into

JB-

respectively, acting parallel

ular to

H causes hand,

and perpendic-

EF.

The

force

move along EF; the force V, on the other would cause it to move vertically downwards, but the a to

downward push

of a against the guide c develops a reaction is equal and opposite to V. Hence,

in the guide c that

R V

R

are balanced, the net vertical force is zero, and as a In this way, any result there can be no motion vertically.

and

move in any direction except creates a reaction in one of the restrainalong the line ing bodies &, c that exactly neutralizes that tendency. force that tends to cause a to

EF

3. Definition of a Machine. A machine is an assemblage of .fixed and moving parts so arranged as to utilize energy derived from some external source for the purpose of doing work. In the operation of machinery, motion and force are communicated to one of the movable parts and transmitted to the part that does the work. During the transmission, both the motion and the force are modified in direction and amount, so as to be rendered suitable for the purpose to which they are to be applied.

LINK MECHANISMS

3

The moving

parts are so arranged as to have certain motions relative to each other, the effect of which is the compel part doing the work to move in the required way. The nature of these movements is independent of the definite

to

amount

of force transmitted; in other words, in a

model

of a

machine operated by hand, the relative motions of the parts will be precisely the same as in the machine itself, although

in the latter case a great amount of power mitted and much work done.

may

be trans-

GENERAL KINEMATIC PKINCIPLES KINDS OF CONSTRAINED MOTION Plane Motion. All constrained motions

4.

of rigid however complicated, may be divided into three classes; viz., plane motion, spherical motion, and screw motion. body is said to have plane motion when all its

bodies,

A

points in parallel planes. Nearly all the motions of machine parts belong to this class. For example, the motions of all parts of the steam engine, except the governor, are plane; all points of the piston, piston rod, and crosshead

move

move

equal parallel straight lines;

all

in

points of the crank, shaft,

and flywheel move in circles of various radii that lie in parallel planes; and points of the connecting-rod describe oval curves, which likewise lie in parallel planes. There are two special cases of plane rotation

axis

motion of much importance, namely, and translation. A body is said to rotate about an

when

centers

all

all lie

points of it move in parallel circles whose on the axis; this motion is common.

very Thus, a flywheel rotates about the axis of the shaft, a pulley about the axis of the line shaft, etc. A body has a motion of translation when the direction of a straight line in that

body

is

always parallel to or coincides with its original direction. of any points in the body may be either straight or curved. The motions of the piston of a steam engine and the parallel rod of a locomotive are examples of translation in straight and curved paths, respectively.

The paths

LINK MECHANISMS

4

A body has spherical motion remains always at a definite distance fixed point so as to move in the surface of an imaginary sphere with the fixed point as a center. In machinery, there are few examples of spherical motion; the universal joint and the balls of the steam-engine flyball govSpherical Motion,

5.

when each from some

point of

it

ernor are two familiar examples of this motion.

Screw motion

6.

consists of a rotation about a fixed

combined with a translation along the axis. An example is the motion of a nut on a bolt. Of the three forms of motion, plane motion occurs most frequently in machinery. Unless the contrary is stated, it is assumed in the following pages that the motion of a machine axis

part is plane.

PLANE MOTION OF A RIGID BODY Point Paths.

7.

During the motion of a rigid body, such as a machine part, each point of the body traces a line, which

is

called the

path

of the point. In plane motion, the in the case of a machine part the usually a closed curve; that is, a path that, if fol-

path

lies in a plane,

path

is

lowed continuously, position.

The path

and

will

bring the

of any point of a

body to its original body rotating about

a fixed axis is a circle, which is a closed curve. The direction in which a point is moving at any instant is the direction of the tangent to the path of motion at the given point.

Also, in plane motion, the motions of any two points of a the motion of the body as a whole. Thus,

body determine

in the case of a connecting-rod, if the motions of any two points of the rod, at any instant, are known, the motion of

the entire rod for that instant is determined.

8.

The Instantaneous Center. In Fig. 2, A and B are a rigid figure, which may be in any plane sec-

two points of

tion of the rigid

and of

A

body

parallel to the plane of motion,

and

m

are their paths. The direction of motion at a certain instant is the tangent a, at the point A, to

n, respectively,

LINK MECHANISMS the path m. Similarly, the tangent b to the path n shows the direction of motion of at the same instant. Let the lines e and / be drawn through and perpendicular, to the respectively, tangents a and b, and let O be their inter-

B

A

section.

B

some

Suppose point be chosen, say E, on the e, and that the figure be rotated about this point. For the sake of clearness, imagine the figure to be a disk of paper with a pin stuck through it at the point E. Evidently, when the disk is rotated, the direction of motion of will be that

line

A

AE,

perpendicular to a.

gent cause

A

that

is,

in the direction of the tan-

a rotation about

Hence, move, for the

to

any point on the

line e will

in-

stant, in the direction of the

tangent a. In the same way, a rotation about any point in the line / will cause B to move in the direction of the tan-

gent b. Therefore, by choosing as the center of rotation the point

=

v

- rw

-

r

and

(1)

Formula 1 may be used to determine the angular velocity body rotating about an instantaneous, instead of a fixed, axis, or, what is the same thing, the of a

point

angular velocity of a a plane figure rotating about an instantaneous Thus, in Fig. 3, suppose that the velocity of the is 20 feet per second and the distance from to the

of

point center.

D

D

LINK MECHANISMS

P

is 4 feet; then the angular velocity of instantaneous center the body c about the center of rotation is, by formula 1-

P

^

20 _____ 4 v

-

5

r

velocity of the body, the linear velocity

Having the angular any other point, as

of

Cor

thus,

is

E,

is easily

found by formula 2;

linear velocity of

C

angular velocity

linear velocity of

E

angular velocity

X PC X PE

In the foregoing consideration of angular velocity, the unit based on the angle subtended by an arc equal to the radius

of the circle forming the path of the point in motion; this In other words, a radian is the is called a radian.

angle

angle subtended by an arc equal in length to its radius. The length of the circumference of a circle, that is, the arc subtending the angle of one complete revolution, or 360,

is

2 nr.

fore equal to

one radian

is

This angle, measured in radians,

=

2rc radians.

r equal to

~- =

there-

is

It follows, therefore, that

57.296.

RELATIVE MOTION 12.

Two

bodies,

each of which

some

is

moving

relative to

fixed body, have, in general, rela-

For example, the crank and the connecting-rod of an engine have each a certain motion relative to the frame; each has also a motion relative to the other, which motion is tive motion.

F<

a turning about the axis of a crankpin.

An

illustration

relative

block b

rod FIG. 6 it

a,

motion is

and

is

of

the

shown

principle in Fig. 6.

constrained to this

rod

is

slide

pinned at

A

of

A

on a to a

fixed body, so that the only possible motion of a is a rotation about A. Now,

a does not move, the block

b

simply slides along

a;

if,

LINK MECHANISMS

9

however, a rotates while b slides along it, the actual motion of b will have the direction P G, but its motion relative to a is in the direction P E, just as though a were at rest. That is, the motion of b relative to a is not in the least affected by the motion of a. In general, the relative motion of two bodies is not affected by any motion they may have in common. This is a principle of great importance, and may be further illustrated

by the following familiar examples: The relative motions of the parts of a marine engine are not influenced by

the. rolling of the ship. The relative motions of the moving parts of a locomotive are not affected by the motion of the locomotive on the track.

From

this principle, it follows that in

studying the relative

common to both may be common motion may be given

motions of two bodies, any motion

neglected; also, if desirable, a them without affecting their relative motions.

LEVERS Use of Levers. guide a moving point,

13. to

Levers

mechanisms moving rod, or to

are used in

as the end of a

transfer motion from one line to another.

There are three Levers whose lines of motion are parallel; (2) levers whose lines of motion intersect; and (3) levers having arms whose center lines do not lie in the kinds of levers:

same

(1)

plane.

In proportioning levers, the following points should in general be observed; they apply to all three cases just

mentioned: 1.

When

in mid-position,

the center lines of the arms

should be perpendicular to the lines along which they give or take their motions, so that the lever will vibrate equally each way. 2.

If

a vibrating link

is

connected to the lever,

of attachment should be so located as to

move

its point equally on each

motion of the link. arms must be proportional to the distances through which they are to vibrate. side of the center line of 3.

I

The lengths

LT

374B

2

of the lever

LINK MECHANISMS

10

14.

An

Reversing Levers.

illustrates the foregoing principles

SR

example

shown

is

of a lever that in Fig. 7.

The

RH

the driver, and through the connecting-rod approximately along the center gives a motion to the point to the the lever is transferred

crank line

A JB,

is

H

EH

by

which

lever vibrates equally each way about its cb and da. fulcrum or center 0, as indicated by the lines

line

CD.

When

The

in mid-position, its center line

to the lines of

CD and A B. points E and H,

motion

traversed by the tional to the arms

E

and

EH

is

perpendicular

horizontal distances respectively, are propor-

The

H O, or y

:

E

= x HO. The :

j>

I

|_

^r as

I

M

FIG. 7

E

EK

with the rod KD, connects the point vibrating link line by the which is constrained to move in a straight the lever is so o-uide g> and, in accordance with principle 2, will be as far above the center proportioned that the point as it will be below line of motion CD, when in mid-position, that is, the points c and d are it in the two extreme positions; is above it. as far below the line as the point connects the bottom of the lever, where the rod

E

E

At

HR

H

the point being with the crank, the same principle holds, a and b are above it. as far below the line A B as the points AB and lines the center Frequently, the distance between and the extent of the motion along these lines, is

CD

given,

LINK MECHANISMS

11

from which to proportion the lever. A correct solution to this problem is troublesome by calculation, because it is not

known

at the start

A

how

far

motion the points

lines of

graphic solution

CD

motion

lines of

is

and

dicular to them.

Draw

from

y,

equal to

it

or half the

CD;

along

shown

A B>

ME

Draw

in Fig. 8.

and a center parallel to

line

ST

at

the center

S T perpena distance

stroke

also, the

HN,

parallel line

on

57

the other side of

and

above and below their respective

E and H should be.

1

,

from

at a distance

equal to i x, or half the stroke along B. it

A

Connect points Mand.

N by a where sects

straight line;

this line inter-

6*

O

T, as at

s

be the center or

crum With

of

find

trial

by

the

will ful-

lever.

as a center, the radius

of an arc that will cut

ST as far below the line AB as does it

H N above

this line,

or so that the distance

n

Pie 8 -

will be equal to the distance

m.

As

an aid in determining the correct radius, describe an arc cutting S T, with O as a center and a radius ON. The distance n will be a little more than i /. Now, draw a straight line and O. through points

The

part included

between

H N and ME

H

determines the

length of the lever.

arm

is

equal to

In this case, the length of the shorter E, and that of the longer arm to OH.

15. Noii- Re versing Levers. Fig. 9 shows the same construction applied to a lever in which the center O is at

LINK MECHANISMS

12

one end

of the lever.

This lever does not reverse the motion motion along A B is to CD will be in the same

like the previous one, since, when the the right or left, the motion along

direction.

The

figure

is

lettered like the preceding

one, so that the construction will be easily understood.

16.

tions.

Reducing MoIt is

able that a lever

often desir-

mechanism

on a greater or smaller scale, along one line, the exact motion that shall reproduce

occurs along another line; that

is,

for every

in the rate of the

change motion

along one line a corresponding change shall be produced indialong the other line. Figs. 10 and 11 illustrate three cator reducing motions that accomplish this. In Fig. 10, the lower end of the lever attaches to the cross-

head of the engine through the R. The indiswinging link

H

cator string

bar

fastened to the

is

CD, which

receives

its

mo-

from the lever through the link EK, and slides through the

tion

guides g,g in a direction parallel

AB

to the line of motion

of the crosshead. the bar

In order that

CD shall have the

kind of motion as the

same cross-

it is necessary that the FlG 10 and lengths of the links shall be proportional to their respective lever arms; = OE\ Elf The pins must be so placed thus

head,

HR

EK

OH: HR

'

LINK MECHANISMS

13

that the connecting; links will be parallel; if parallel at one point of the stroke, they will be so at all points. When the links JSKand HJt are = parallel,

the length of the indicator

OK\ OR

diagram

to the length of the stroke as

O:OH,and

will bear the

O E bears

to

same

ratio

OH.

be observed that the pins O, are in one K, and straight line, and, in general, it may be said that any arrangement of the lever that will keep these three pins in a straight line for all points of the stroke will be a correct one. In Fig. 11, two such arrangements are shown. In the first, the pins and R are fast to the slide and

R

It is to

K

crosshead,

FIG. 11

respectively, and slide in slots in the lever. In the second, they are fast to the lever, the slots being in the slide and crosshead. In both, the pins and are in a straight line with the pin O during the whole stroke.

K

17.

R

Bell-Crank Levers.

Levers whose lines of motion termed bell-crank levers; they are used very extensively in machine construction for changing the direction and the amount of motion. The method of" intersect are

laying out

a lever of this kind to suit a given condition is as follows: In Fig. 12, suppose the angle CAB, made by the lines of motion, to be given, and that the motion along is to be

AB

LINK MECHANISMS

14 twice that along

A

distance from

AB

Draw c d

C.

venient distance from

parallel to

A C. Draw a b

AC

parallel to

equal to twice the distance

at

any con-

A B and at a

otcd from

A C.

FIG. 12

Through

the intersection of these

of the angle,

crank

may

draw the

line

A F.

two

The

be taken at any point on

lines

and the apex

center

A F suited

(9

A

of the bell-

to the design

FIG. 13

of the machine. Having chosen point (9, draw the perpendiculars C^and OH, which will be the center lines of the

lever arms.

LINK MECHANISMS In Fig. 13, a construction when the two lines and

CD

limits of the drawing.

is

shown

AB

In Fig.

that

15

may be employed

do not intersect within the the same construction is

14,

PIG. 14

applied to a non-reversing lever, in which the center O falls outside of the lines A and CD. The figures are lettered alike, and the following explanation applies to both. Draw cd parallel to and a b

B

CD

A

parallel to

B, as before, so that the distance of cd

from from

CD

:

distance of a b

A B = amount of motion along CD amount of motion along A B. Again, :

draw

lines

gh

and ef

in

exactly the same way, but taking care to get their

distances from

CD and A B

different from those of the lines just drawn. Thus, if

cd should be

6 inches

FIG. 15

from CD, make gh some other distance, as 4 inches or 8 inches, and then draw .

As

LINK MECHANISMS

22

Harmonic motion may be defined

as the motion of the foot

of a perpendicular let fall on the diameter of a circle from a the circumference. point moving with uniform velocity along

The Toggle joint.

23.

The togglejoint, shown

in

heavy pressure by Fig. 20, is a. mechanism for producing It will be seen that it the application of a small effort. resembles the steam-engine mechanism with crank and cona

F

is The effort of about the same length. at the slide. applied at the joint B and the resistance The mechanical advantage, that is, the ratio of resistance It is to effort, depends on the angle between the links. evident that the point Q is the instantaneous center of

necting-rod

P

motion of the link

\-~* o

_

ri-

1H'

Ag

Now>

angle

that

smaller,

the

the

ag

ABQ becomes is,

as

links

approach the position in which they would form a straight horizontal FIG 20

line,

.

moves toward that of

B

Since

the joint

grows

Q

is

the

A, and the velocity of

A

center

Q

relative to

less.

the

instantaneous

pendicular to the direction is the moment of

of

center

the

and

force

/*,

QA the

is

per-

product

F

P about Q. about Q is FX. QC. Considering the mechanism to be, for the instant, in equilibrium, and neglecting friction,

PX QA

The moment

of

PX QA

= FX QC,

or (1)

This formula may be used to calculate the pressure P when the force acts vertically at B. If the force acts at some other point than B, as, for example, F' at E, it becomes necessary to find the equivaare lent pressure at the joint B. Assuming that F' and

F

parallel,

and taking moments about O, F'

X O D = F'X O

G,

LINK MECHANISMS

p=

or

j

pt

0^

and

OG'

TW OD

this value of

=

'

are similar.

SmCe

~0~G

~O&

Hence,

F= F> X

fin formula

23

Substituting

||.

1,

OBY.QA In case the force acts other than vertically, it is necessary find its vertical component, which, when substituted for ForF' the proper formula, will give the value of />. to

m

The

distances

QC and QA

tion of the toggle.

accurately

to a

will

vary according to the posi-

However, by laying out the mechanism

fairly large

scale, these

distances

may be

measured with sufficient accuracy, so that the pressure be calculated for any. position.

*AMJC i.-m Fig. 20,

let

^equal

5 inches; find the pressure P.

,

SOLUTION.

L

Q

.

2 '~ lQ a

^is 32 QA, 6 inches;

80

X

24 ~

=

384

like

t0gglejoint

Ib.

that

Ans.

shown

in

-ce

d

what

19

2

ln

v

"

=

the SS>

substituting the values given,

400

lb.

Ans.

EXAMPLES FOR PRACTICE ^^ is 8 inches and Q C is 36 inches,

if

be applied vertically at

and

'

the pressure P?'

~

In Fig. 20,

Pig. 20,

'

is

SoLimoN.-Applying formula 2 and

4

/may

inches- and C, 24 mcnes,

Applying formula 1,

P= X

80 pounds;

B to

P of

produce a pressure

what force 180 pounds? Ans. 40 Ib.

In a togglejoint similar to that in Fig. 20, the arms OB, equal, each being 14 inches in length; the height and the force is 2,800 pounds. Find the equais

BQ are

AH

B G*

F

inches;

CQ

Ans. 4,695.74

QC

>

24 inches

!

^^-

Ib

16

Ans. 462

Ib.

LINK MECHANISMS

24

F

of 200 pounds, Fig. 20, should act at right In case a force 4. when Q C equals 32 inches; at JB, find the pressure angles to the angle that P QA, 8 inches; and OQ, 60 inches, remembering that thp to is ingle Q O A. makes with the vertical equal Ans. 793 lb., nearly

P

OB

A point revolves about an axis at a speed of 4,200 feet per 5. what is its angular velocity minute; if the point is 5 feet from the axis, Ans. 14 radians per sec. in radians per second? The angular velocity of a point is 25 radians per second and its 6. distance from the axis of revolution

is

8 feet;

what

is its

linear velocity?

Ans. 200

per sec.

ft.

of 24 feet makes 56 revoflywheel having an outside diameter in radians lutions per minute; find the angular velocity of the wheel, 5 86 radians per sec. Ans per second. 7.

A

-

-

QUICK-RETURN MOTIONS 24.

Vibrating-Link Motion.

motions J

Quick-return

are used in shapers, Blotters, and other machines, where all the useful

JJ LL

s

-

~

work

done during

is

the stroke of a recip-

rocating piece in one direction. During the

working tool

the

stroke,

must move

at

a

suitable cutting speed,

while stroke, is

on

performed,

sirable

return

the

when no work that

it is it

de-

should

travel more rapidly.

The mechanism shown in Fig. 21, known as a vibrating-link motion, is applied to

shaping machines operating on metal. Motion is received from the pinion p

FlGl21

t

which drives the gear g.

The

pin b

is

fast to the gear,

and

LINK MECHANISMS pivoted to

it is

and

CD

the block

which

25

is fitted to

the slot of the oscillating link CD. As the gear rotates, the pin describes the circle bedc, the block slides in the slot of the link /,

CD

causes to oscillate about the point D, as indicated by the arc CC', the path of the joint The rod / connects the upper end of the link with the .tool slide, or ram, r, which is constrained by guides (not shown) to reciprocate in a straight horizontal

C

line.

During the cutting stroke, the pin b travels over the arc deb, or around the greater arc included between the points of tangency of the center lines and CD During the return stroke, the pin passes over the shorter arc bed and as the gear f rotates with a

CD

uniform

of the forward

throw of the

velocity, the time, will be to each other as tha to the length of the arc bed The

and return strokes

length of the arc

deb

is

slotted link and the travel of the tool can be varied by the screw s, which moves the block / to and from the center of the gear. The rod /, instead of vibrating equally above and below a center line of motion is so arranged that the force moving the ram during the cutting stroke will always be downwards, causing it to rest firmly y on the guides.

25. To lay out the motion, proceed as follows: Draw he center line S T, Fig. 22, and parallel to it the line mn he distance between the two being equal to one-half the ongest stroke of the tool. About 0, which is assumed to be the center of the gear, describe the circle bdc with a radius equal to the distance from the center of ,

the center of the gear #, Fig. 21,

the pin b to

when

set for the longest Divide the circumference of the circle at b and d into upper and lower arcs extending equally on each side of the center and such that their ratio is equal to that of linear, the times of the forward and return strokes. In this case the time of the forward stroke is double that of the return stroke and the circle is divided into three equal as

stroke.

parts, shown at 6,d, and c, thus making the arc db equal to one-half the arc deb. Draw the radial lines Ob and O d. I

L T 374B-3

Through

b

LINK MECHANISMS Z?

where

point

.eve. . the upper end of the slotted

c

,tt

draw the

Draw

horizontal line

CO, making

C'A which will be tangent the other extreme position

to

it

interit

Cohereh C

C'-E equal to

the

orde

at

rf.

CA. thus

of the lever.

FIG. 22

on the of intersection

h

L and ineoyn the line

R

numoei

on

me

ate c

,.

LU

c

IL

will intersections so that they the -

LINK MECHANISMS point 2 to

27

on the forward stroke. On the return stroke, from point 9 to 10, 11, 12, and 1, the motion is much 3,

etc.,

less uniform.

A property of this motion is that, as the radius Ob is diminished to shorten the stroke, the return becomes less rapid, as can be seen from the figure motion when

Ob

the radius with the

is

by comparing the motion when Ob' is

the radius.

26. anism

Whitworth Quick-Return Motion. is

the

in

shown

side

of

This mech-

in principle in Fig-. 23.

the

gear

/,

The pin d, inserted gives motion to the slotted

8

PIG. 23

link

CD,

as

the vibrating link motion. This motion closely resembles the previous one, the difference being that the center of the slotted link lies within the circle described by the pin 6, while in the previous case it lies without it in

D

To

accomplish this result, a pin^

is provided for the gear large enough to include another pin placed eccentrically within it, which acts as the center for the link CD. With this arrangement, the slotted link, instead of oscillating, follows the crankpin during the complete revolution, and thus becomes a crank. The stroke line passes through the center D, which is below the center O of the pin p. The forward or working stroke occurs

to

turn on, and

D

RL

is

made

LINK MECHANISMS

28

-

the shorter pin describes that n

r fr P IIILU

uvvw

t*ivz -velocity

ratio,

order that toothed wheels the

common normal

may

to the

have a constant

tooth curves

.always pass through a fixed point on the line of

8.

Pitch. Surfaces

the circles e and

/,

and Pitch lanes.

with centers

A

must

centers.

In Fig. 8 let

and B, be the outlines two rolling cylinders

of in

Let the tooth curves m and n be so constructed that their common normal

contact at P.

shall always pass through P. The velocity ratio produced by the rolling of these cylinders is precisely and n\ for in the case as that caused by the teeth of rolling cylinders the velocities are inversely proportional the

m

same

to the radii, that is,

V,

N

b

AP

GEARING

7

The bounding

surfaces e and / are called pitch surfaces, and the lines e and / are called pitch lines or pitch

curves,

in the case of circular gears,

pitch circles. tangency P of the pitch lines is called the pitch point, and it must lie on the line of centers. This is the meaning of the term as generally used by designers, and is the meaning intended wherever the term pitch point is mentioned in this treatise on gearing. In the machine shop, however, this term is frequently used in a different sense, being there considered as any point in which the tooth out-

The

or,

point of

line intersects the pitch circle, as indicated in Fig. 9.

P

In the case of circular pitch^lines, the pitch point lies but if the pitch lines are non-circular, as in the case of rolling ellipses, Fig. 4, the pitch point

in a fixed position;

moves along

the line of centers.

circular wheels, the

law of Art. 7

In order to include nonbe made general, as

may

follows:

Law. shall

In order that the motion prod^lced by tooth driving the rolling of two pitch surfaces, the

be equivalent to

common normal through

to

the pitch

the tooth curves

must at

all

times pass

point.

The

object, then, in designing the teeth of gear-wheels is shape them that the motion transmitted will be exactly the same as with a corresponding pair of wheels or cylinders without teeth, which run in contact without slipping. In to so

two general systems of gear-teeth are used. as the involute system and the other as the system, both of which will be discussed in succeeding

actual work,

The one cycloidal

is

known

paragraphs. 9. Definitions. Referring to Fig. 9, which shows part of a circular gear-wheel, the following definitions apply to the lines and parts of the tooth.

The

the outer ends of the teeth

is

addendum

the spaces circular

drawn through

circle

called the

is

circle; that drawn at the bottoms of called the root circle. In the case of non-

these would be called, line and the root line.

gears,

addendum

respectively,

the

GEARING The addendum

is

the distance between the pitch circle

The a radial line. and the addendum circle, measured along circle and the root root is the distance between the pitch The term addendum is a radial line. circle, measured along also

of applied to that portion

frequently

to

a tooth lying

and the term root between the pitch and that portion of the tooth lying

between the pitch and addendum

circles,

root circles.

FIG. 9

addendum, that is, the part of is called the the working surface outside of the pitch circle, root is called face of the tooth. The working surface of the

The working

the

flank

surface of the

of the tooth.

The diameter

of the pitch circle

is

called the pitch

diam-

word diameter is applied to gears, it is diameter unless otheralways understood to mean the pitch diameter wise specially stated as outside diameter, or at the root. eter.

When

the

The distance from a point on one tooth to a correspondingpoint on the next tooth, measured along- the pitch circle, is the circular pitch.

The fillet is a curve of small radius joining- the flank of the tooth with the root circle, thus avoiding the weakening effect of a sharp corner.

THE INVOLUTE SYSTEM

'

10. Production of the Involute Curve. In general, the involute of any curve may be defined as the curve that is

described by a point in a cord as

it is

unwound from

the

original

curve, keeping the unwound portion of the cord straight. Thus, suppose that a cord is wound around the

P

curve a, Fig. 10, and let be any point on the cord. Then, as the cord is unwound, the point

m

P

will describe a curve

FIG. 10

an involute of the curve Pi A being the last position of the cord shown. In the case of an involute to a circle, it is convenient to conceive the curve described as follows: Suppose a, that

is

Fig. 11,

FIG. 11

to be a circular pulley

having a cord wound around it, and let the pulley be pinned at a fixed At the end P point 0. cf the cord a pencil is attached, and there is a fixed groove

GEARING

10

Suppose also that the or guide e tangent to the pulley at P. or cardboard c. pulley has attached to it a sheet of paper take hold of the pencil and pull it along the groove e. rotate about O, pulley and paper will thus be caused to and the pencil will trace on the moving paper the involute so that Pi t or m. If now the pulley is turned backwards,

Now The

P

,

P

is wound up, the pencil will move from P, to the moving paper, the s i on retrace the curve

the string

and

will

PP

point

P

l

moving

to P, its

original position, Let t be a tangent to

m

the curve

From

P,.

at the

the

which the curve duced,

it

point

manner

in

is

prois evident that

the tangent is perpendicular to P*. Similarly, the tangent t t at any other

P

and is is the common normal of the tooth curves, line teeth. therefore the line of action of the pressure between the The angle between the common normal to the tooth surfaces

the

pitch

A A'

circles

.

ED

and the

tangent to the pitch circles is called the In the case of involute teeth, this

common

angle of obliquity. angle

is

constant.

13. Angle and Arc of Action. The angle through which a wheel turns from, the time when one of its teeth comes in contact with a tooth of the other wheel until the line of centers is the angle point of contact has reached the of approach; the angle through which it turns from the instant the point of contact leaves the line of centers until the teeth are no longer in contact is the angle of recess.

The sum The arcs are

of these

called

of the

the

two angles forms the angle of action. measure these angles recess, and action,

pitch circles that arcs of approach,

respectively. In order that one pair of teeth shall be in contact until the next pair begin to act, the arc of action must be at least

equal to the circular pitch. The path of contact is the line described by the point of In the case of involute gears contact of two engaging teeth. the path of contact is part of the common tangent to the base

GEARING circles.

The

amounts

to the

13

arc of action depends on the addendum, or same thing, on the length of the tooth.

what With

short teeth, the arc of action must necessarily be small; and a long arc is desired, the teeth must be made long.

if

14. Standard Interchangeable Gears. In order that two gears may run properly together, two conditions must be satisfied: (1) They must have the same circular pitch, and (2) they must have the same obliquity. If, therefore, all involute gears were made of the same obliquity, any pair of wheels having teeth of the same pitch would work properly together, and such gears would be said to be interchangeable. The tooth selected for the standard is one having an angle of obliquity of

that

15;

is, in

Fig. 12, angle

T CE

angle

CO' E = 15. With this obliquity, then, in the triangle 0' EC, 0' E = O'C cos CO E = O1 C cos 15 = .966 O> C; that is, 1

the radius of the base circle equals .966 times the radius of

The distance between the base circle and the pitch circle. pitch circle is thus about one-sixtieth of the pitch diameter. In the interchangeable series of standard gears, the smallest

number of teeth number the

smaller

that a gear

may have

arc of contact will

is

twelve, for with a

be smaller than the

cir-

cular pitch, in which case one pair of teeth will separate before the next pair comes into contact, and the gears will not run.

The Involute Hack. -A rack

15.

is a

series of gear-

teeth described on a

pitch line. usually a metal

straight It is

in which the teeth are cut, although they may be

bar,

A rack,

cast.

fore,

may be

there-

E

consid-

ered as a portion of the circumference of a

gear-wheel whose

long, and whose sequentlv be regarded as straight.

radius

is

infinitely

FIG. IS

pitch line

may

con-

GEARING

14

the sides of the teeth are In the involute rack, Fig. 13, - 15 = 75 with the making an angle of 90 on the contact side, the tooth outlines Thus, EF. line pitch N. To avoid interline of action are perpendicular to the ends of the teeth should be rounded to run with straight lines

N

'

ference, the

the 12-tooth pinion.

with

a

A

pinion

a

is

small gear meshing

rack or with a larger gear.

16.

An annular, or Internal Gears. one having teeth cut on the inside of The pitch circles of an annular gear and its pinion

Involute

internal, gear the rim.

is

have internal contact, as shown

in Fig. 5.

FIG. 14

The construction of an annular gear with involute teeth is shown in Fig. 14. The obliquity of 15 is shown by the TCN, and the base circles EE' and D D are drawn 1

angle

tangent

to the line of action

ively, as centers.

NN'

The addendum

t

with

and

O', respect-

circle for the internal

gear

should be drawn through F, the intersection of the path of drawn from the contact NN' with the perpendicular The teeth will then be nearly or quite center of the pinion.

OF

without faces, and the teeth of the pinion, to correspond,

GEARING may be

without flanks.

15

If the two wheels are nearly of the will interfere; this interference may

points c and d be avoided by rounding the corners of the teeth.

same

size,

THE CYCLOIDAL OR ROLLED-CURVE SYSTEM The name cycloid is given to the Tlie Cycloid. curve traced by a point on the circumference of a circle as it 17.

rolls

on

the line

P

point

a straight line.

Thus,

in Fig. 15 the circle

m rolls on

A

B, and the on the cir-

cumference traces the path A CB, which is a cycloid.

ing the

circle

The is

roll-

called .

generating cir-

cle, the point

P

the

15

tracing point, and the

line

AB

the

base line. If

the base line

AB is

an arc of a

circle, the

curve traced

by the point P is called an epicycloid when the generating circle rolls on the convex side, and a hypocycloid when it rolls on the concave side. Fig. 16 shows an epicycloid, and Fig. 17 a hypocycloid. of

The one property cycloidal curves

that

makes them

peculiarly suitable for tooth outlines is that

the

common normal

through the point of contact will always pass through the pitch

Pm

'

point.

16

16,

and

In 17,

Figs.

P

is

15,

the

E

is the point of contact of the tracing point in each case and generating circle and the line on which it rolls. Now, it is

clear that the generating circle

about

E as

m

a center, and the pQfflt

is

P

for the instant turning therefore moving in a

is

GEARING

16

PE.

direction at right angles to

curve at

to the

P is

In other words, the tangent

perpendicular to

PE,

or

PE

normal

is

to the curve.

A

particular

form of

the i*

hypocycloid is shown in Fig. 18. The circle

generating

m has

a diameter equal to the radius of the circle n.

In this case, a point

P of m

describes a straight line

ACB,

which passes through the center C of n and is therefore a diameter.

18. Generation of Tooth Outlines. In Fig. 19 let a be two pitch circles in and contact at E and suppose a third circle m to be in contact with them at E. If the circle m rolls on the outside of the !

FIG. 18

P

describes the epicycloid describes the rolls on the inside of circle a

circle a, the tracing point

P

If, however, it B be hypocycloid C P d. Now let an arc of the curve B taken as the outline of the face of a tooth on a, and let an arc be taken as the outline of the flank of the hypocycloid l

,

P

CPd

of a tooth

on

PE

is

over,

These curves are in contact at P, and, morenormal to each of them. Hence, the common

a-,.

normal through the point of contact passes through the pitch point, and the curves satisfy the condition required of tooth outlines.

The one

rolling circle

m

generates the faces of the teeth of

and the flanks of the teeth of a*. A second circle m it rolling on the inside of a and on the outside of a^ may be used to

a

generate the faces of gear-teeth for

a^

and the

flanks for a.

While not necessary, it is customary that the generating circles shall have the same diameter. If a series of gears of the same pitch have their tooth outlines all generated by rolling

GEARING circles of the

17

same diameter, any gear will run with any other

gear of the series; that

is,

the gears are interchangeable.

PIG. 19

19.

Size of Generating Circle.

shown

In Figs. 20 to 22

is

the effect of different sizes of

generating circles on the flanks of

FIG. 21

FIG. 20

the teeth.

In the

first,

FIG. 22

the generating circle has a diameter

GEARING

18

equal to the radius of the pitch circle, the hypocycloid is a straight line, and the flanks described are radial. In the second, with a smaller circle, the flanks curve away from the radius, giving a strong tooth, and in the third, with a larger circle, the flanks curve inwards, giving a weak difficult to cut. It would seem, therefore, diameter for the generating circle would be one-half the pitch diameter of the smallest wheel of the set, or one-half the diameter of a 12-tooth pinion, which, by common consent, is taken as the smallest wheel of any set. It has been found, however, that a circle of five-eighths the diameter of the pitch circle will give flanks nearly parallel, so that teeth described with this circle can be cut

and one

tooth,

that a suitable

For this reason, some gear-cutters with a milling cutter. are made to cut teeth based on a generating circle of fiveeighths the diameter of a 12-tooth pinion, or one-half the

diameter of a 15-tooth pinion. It is

more common

practice to take the diameter of the

generating circle equal to one-half the diameter of a 12tooth pinion rather than one-half the diameter of a 15-tooth pinion; this size, therefore, is taken in this discussion of the

subject of gearing.

20.

Obliquity of Action.

Neglecting

friction,

the

pressure between two gear-teeth always has the direction of the common normal. In the case of involute teeth, this

always the same, being the line of action ED, teeth, however, the direction ot the constantly changing, and hence the direction of

direction

is

Fig. 12.

With cycloidal

normal

is

pressure between the teeth is likewise variable. In Fig. 19, let the tooth curves be just coming into contact at the point P.

The

that instant is

direction of the pressure between them at since is the common normal at P.

PE

PE,

'

gears rotate, however, the point of contact P approaches E, moving along the arc PPi E, and the common

As

the

normal

when

PE,

P

to 0, 0.

therefore,

makes

reaches E, the

a greater angle with O^ 0, until,

common normal

is

at right angles

GEARING

19

Beyond the point E, the point of contact follows the arc until the teeth leave contact, and the direction of the

EQ

common

tangent gradually changes from a position at right to the position EQ. angles to O l Hence, with cycloidal teeth, the obliquity of the tooth

the teeth point,

first

come

in contact,

and increases again

maximum at the final point a rule, the greatest obliquity should not

As

of contact.

pressure is greatest when decreases to zero at the pitch

to a

exceed 30.

21. Rack and Wheel. -In Fig. 23 and 12-tooth pinion. Tooth outlines for the

is

shown a rack

rack in the cycloidal system are obtained

by rolling the generating circles oo' on the straight pitch line the curves are

B B;

therefore cycloids. The generating circle o

passes through

the center

O

of the

pinion and rolls the pitch circle

on

A A,

22.

FIG. 23

and therefore describes radial

flanks.

Epicycloidal Annular Gears.

Annular gears, or internal gears, as already explained, are those having teeth cut on the inside of the rim. The width of of an space

internal gear is the

gear.

Two

same

as the width of tooth of a spur generating circles are used, as before, and if

they are of equal diameter, the gear will interchange with spur wheels for which the same generating circles are used. In Fig. 24 is shown an internal gear with pitch circle A

A,

inside of

which

B

the pinion with pitch circle B. The generating circle o, rolling inside of B, will describe the flanks of the teeth for the pinion, and rolling inside of A, is

B

A

the faces of the teeth, for the annular wheel. Similarly, the corresponding faces and flanks will be described by o'. The

GEARING

20

in regard to epicycloidal only special rule to be observed internal gears is that the difference between the diameters

FIG. 24

sum

of the pitch circles must be at least as great as the the diameters of the generating circles.

of

is illustrated by Fig. 25. the pitch circle of an internal gear and b that of the

This

a

is

Then,

pinion.

for correct ac-

tion, the difference

D

d of the

diameters must be at least as great as c, the sum of the diameters of the generating circles.

To

take a limiting case, supto have 36 teeth and b

pose a

24 teeth.

A wheel with a diam-

eter equal to FlG- 25

dotted

at